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In this paper we consider j-tuple-connected components in random k-uniform hypergraphs (the j-tuple-connectedness relation can be defined by letting two j-sets be connected if they lie in a common edge and considering the transitive closure; the case j = 1 corresponds to the common notion of vertex-connectedness). We show that the existence of a j-tuple-connected component containing Θ(nj) j-sets undergoes a phase transition and show that the threshold occurs at edge probability

$$\frac{(k-j)!}{\binom{k}{j}-1}n^{j-k}.$$

Our proof extends the recent short proof for the graph case by Krivelevich and Sudakov, which makes use of a depth-first search to reveal the edges of a random graph.

Our main original contribution is a bounded degree lemma, which controls the structure of the component grown in the search process.

A celebrated result of Rödl and Ruciński states that for every graph
$F$
, which is not a forest of stars and paths of length 3, and fixed number of colours
$r\geqslant 2$
there exist positive constants
$c,C$
such that for
$p\leqslant cn^{-1/m_{2}(F)}$
the probability that every colouring of the edges of the random graph
$G(n,p)$
contains a monochromatic copy of
$F$
is
$o(1)$
(the ‘0-statement’), while for
$p\geqslant Cn^{-1/m_{2}(F)}$
it is
$1-o(1)$
(the ‘1-statement’). Here
$m_{2}(F)$
denotes the 2-density of
$F$
. On the other hand, the case where
$F$
is a forest of stars has a coarse threshold which is determined by the appearance of a certain small subgraph in
$G(n,p)$
. Recently, the natural extension of the 1-statement of this theorem to
$k$
-uniform hypergraphs was proved by Conlon and Gowers and, independently, by Friedgut, Rödl and Schacht. In particular, they showed an upper bound of order
$n^{-1/m_{k}(F)}$
for the 1-statement, where
$m_{k}(F)$
denotes the
$k$
-density of
$F$
. Similarly as in the graph case, it is known that the threshold for star-like hypergraphs is given by the appearance of small subgraphs. In this paper we show that another type of threshold exists if
$k\geqslant 4$
: there are
$k$
-uniform hypergraphs for which the threshold is determined by the asymmetric Ramsey problem in which a different hypergraph has to be avoided in each colour class. Along the way we obtain a general bound on the 1-statement for asymmetric Ramsey properties in random hypergraphs. This extends the work of Kohayakawa and Kreuter, and of Kohayakawa, Schacht and Spöhel who showed a similar result in the graph case. We prove the corresponding 0-statement for hypergraphs satisfying certain balancedness conditions.

A uniform hypergraph H is called k-Ramsey for a hypergraph F if, no matter how one colours the edges of H with k colours, there is always a monochromatic copy of F. We say that H is k-Ramsey-minimal for F if H is k-Ramsey for F but every proper subhypergraph of H is not. Burr, Erdős and Lovasz studied various parameters of Ramsey-minimal graphs. In this paper we initiate the study of minimum degrees and codegrees of Ramsey-minimal 3-uniform hypergraphs. We show that the smallest minimum vertex degree over all k-Ramsey-minimal 3-uniform hypergraphs for Kt(3) is exponential in some polynomial in k and t. We also study the smallest possible minimum codegree over 2-Ramsey-minimal 3-uniform hypergraphs.

For k-uniform hypergraphs F and H and an integer r, let cr,F(H) denote the number of r-colourings of the set of hyperedges of H with no monochromatic copy of F, and let , where the maximum runs over all k-uniform hypergraphs on n vertices. Moreover, let ex(n,F) be the usual extremal or Turán function, i.e., the maximum number of hyperedges of an n-vertex k-uniform hypergraph which contains no copy of F.

For complete graphs F = Kℓ and r = 2, Erdős and Rothschild conjectured that c2,Kℓ(n) = 2ex(n,Kℓ). This conjecture was proved by Yuster for ℓ = 3 and by Alon, Balogh, Keevash and Sudakov for arbitrary ℓ. In this paper, we consider the question for hypergraphs and show that, in the special case when F is the Fano plane and r = 2 or 3, then cr,F(n) = rex(n,F), while cr,F(n) ≫ rex(n,F) for r ≥ 4.

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